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Development of response models for optimising HPLC methods

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Development of response models for optimising HPLC methods
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  Development of response models for optimising HPLC methods W. Dewe´ a, *, R.D. Marini  b , P. Chiap  b , Ph. Hubert   b , J. Crommen  b , B. Boulanger  a  a   Lilly Development Center, Eli Lilly and Company, Rue Granbonpre´ 11, B-1348 Mont-Saint-Guibert, Belgium  b  Analytical Pharmaceutical Chemistry, Department of Pharmacy, University of Lie`ge, CHU, B36, B-4000 Lie`ge, Belgium Received 4 December 2003; received in revised form 24 January 2004; accepted 21 April 2004Available online 21 August 2004 Abstract For optimizing a liquid chromatographic method, direct modeling some responses like minimum resolution is sometimes difficult to perform. By modeling the retention time at apex and the times associated to 13.4% of the peak height, it is possible to obtain estimations of these responses using an adequate cascade of equations, these equations being the definitions of these responses. As a multi-criteria decisionhas to be taken in most of these optimizations, a Derringer’s function can be defined for each response of interest and a overall desirability, onwhich the final is taken, can be derived. In addition, the chromatogram in any experimental point can be predicted. D  2004 Elsevier B.V. All rights reserved.  Keywords:  Liquid chromatography; Retention time; Optimization; Modeling; Multi-criteria decision 1. Introduction The purpose of this paper is to propose a methodology tomodel chromatographic responses during the optimizationof an HPLC method.As an HPLC method allows the analyst to separate thecompounds of a mixture, it can have different applications.For example, if an HPLC method is able to separate, it could be used to identify a compound in the medium. AnHPLC method could also be used in the context of estimating the quantity of a compound in a medium. Suchmethods are mostly used in process chemistry, in pharmaceutical product development and also in drugdisposition.An analyst has to optimize the HPLC method of interest and has to do that with respect to its application. In anycase, this analyst has to make sure that the compound of interest is well separated from the others in a reasonabletime. To illustrate, as most of the quantification is based on peak area, the analyst cannot estimate accurately thequantity of a compound if it is (partly) confounded withanother one.An analyst optimizes an HPLC method by finding the best experimental conditions in order to separate thecompounds of a mixture. First, the analyst determinesthe experimental domain, i.e. the factors (composition of the mobile phase, type of column, pH of mobile phase,etc.) that could influence the separation. In addition, theanalyst selects the response(s) (resolution, retention time,etc.) that will be used to take the decisions. After takinginto account the response assumptions and the factor constraints, the analyst determines the response model(s).Consequently, the analyst builds the experimental designneeded to be able to fit the model(s). Finally, once theexperiences are realized and the model(s) is (are) fitted, theanalyst is able to find this best configuration within theexperimental domain.Some responses may be difficult to model in a direct mode. Indeed, they may vary in such specific manners that cannot be described by the usual modeling techniques. For example, most of the chromatographic responses are either ratios between random variables, or minimum or maximumof a set a random variables. For the resolution, in caseswhere the peaks are narrow, then the time difference on 0169-7439/$ - see front matter   D  2004 Elsevier B.V. All rights reserved.doi:10.1016/j.chemolab.2004.04.016* Corresponding author. Tel.: +32 10 47 64 32; fax: +32 10 47 64 75.  E-mail address:  w.dewe@lilly.com (W. Dewe´).Chemometrics and Intelligent Laboratory Systems 74 (2004) 263–268www.elsevier.com/locate/chemolab  numerator is divided by a small value and consequently, asmall variation on denominator can cause a large variationin the ratio. Another typical example is the minimumresolution that can decrease rapidly to zero when a peak inversion occurs and this is extremely difficult to model in adirect mode. As all the responses have to be modeledanyway to optimize the method, this paper contains a proposal to model in an indirect mode. The paper describesthe modeling of retention factors in a direct mode and thenthe derivation of any other chromatographic response in acoherent manner. 2. Methodology Let us consider a liquid chromatographic method that isunder optimization. The following chromatographicresponses could be used to select the optimal conditions:the peak width, the resolution between peaks, the minimumresolution, the run time (or the maximum retention time),the symmetry factor, the maximum symmetry factor. Let ussuppose a set of factors,  X  1 ,  . . . ,  X   p  on which the methodoptimization will be based. 2.1. Modeling step 1 The first modeling step consists in modeling different retention times associated to each of the chromatogram. Todo so, three retention times will be considered for each peak:the retention time at apex  t  r   and both times associated to  X  %of peak height   t  rl  and  t  rr   ( t  rl b t  rr  ). The retention factors will be modeled instead of the retention times. These retentionfactors are defined as following: k   V  ¼  t  r     t  0 t  0 k  l  V  ¼  t  rl    t  0 t  0 k  r   V  ¼  t  rr     t  0 t  0 where  t  0  is the dead time associated to the column.The peak height considered to obtain the retention times t  rl  and  t  rr   depends on the reference document. If we use theEuropean Pharmacopoeia [1], we will consider a peak  height of 50%. However, if we use the US Pharmacopoeia[2], we will consider a peak height of 13.4%.According to Schoenmakers [3], the retention factor   k   V associated to a retention time at apex has a lognormaldistribution. Consequently, this retention factor has to betransformed by a logarithmic function before any model-ing. Then it is modeled by a multiple regression [4]including the main effects of the factors. This regressionmodel may also contain, if the experimental design isappropriate, the two-by-two interactions between thesefactors as well as the quadratic terms for the continuousfactors.In case of   p  factors, the multiple regression modelincluding main effects, quadratic terms and interactions(sometimes called response surface model) is described asfollows:log  k   V ð Þ ¼  b 0  þ X  pi ¼ 1 b i  X  i  þ X  p j  ¼ 1 X  pk  ¼ 1 b  jk   X   j   X  k   þ  e Regarding this model, a quadratic term appears in thedouble sum sign term when  j   is equal to  k  . When  j   is not equal to  k  , it corresponds to the interaction between thefactor   j   and the factor   k  .In a general manner, a matrix notation will be used for this type of regression model:log  k   V ð Þ ¼  b T   X   þ  e ;  X   being a matrix containing the main effects, quadraticterms and interactions if any and  b T   being a vector containing the coefficients.This regression model allows us to have for the retentionfactor a response surface that can be easily graphicallyrepresented.In addition to the retention factor   k   V , the retention factors k  l  V  and  k  r   V  are modeled in a similar way, as they areassociated to other retention times:log  k  l  V ð Þ ¼  b T  l  X   þ  e l log  k  r   V ð Þ ¼  b T  r   X   þ  e r  Once the models are fitted in the experimental domain,i.e. once the parameters  b T  are estimated, each time isestimated as follows:ˆ t t  r   ¼  t  0  exp ˆ bb T   X    þ  1  ˆ t t  rl  ¼  t  0  exp ˆ bb T  l  X    þ  1  ˆ t t  rr   ¼  t  0  exp ˆ bb T  r   X    þ  1  b T   being the notation for the parameter estimates.Consequently, a response surface is also available for each of these time points.In the cases where the dead time is constant whatever theexperimental conditions, the retention times can be modeleddirectly without considering the retention factors. This is the W. Dewe´ et al. / Chemometrics and Intelligent Laboratory Systems 74 (2004) 263–268 264  case when the same column is used during the optimizationexperiments. 2.2. Modeling step 2 The second modeling step is based on the fact that all thechromatographic responses are functions of the threeretention times defined previously. Consequently, theestimation of the different chromatographic responses isobtained by replacing in their  def inition the retention times by their respective estimation [5].Let us consider first the peak width. As it is defined bythe difference between both times associated to 13.4% of  peak height, the peak width is estimated as following:ˆ dd  ¼  ˆ t t  rr     ˆ t t  rl  ¼  t  0  exp ˆ bb T  r   X   þ  1     t  0  exp ˆ bb T  l  X   þ  1   So at this step, all the peak widths have been estimatedand consequently, the resolution between peaks A and B iscalculated as following: ˆ  R R S ; AB  ¼ 1 : 18ˆ t t  r  ; A    ˆ t t  r  ; B ˆ dd A  þ  ˆ dd B if   t  rl  and  t  rr   are associated to a peak height of 50%2ˆ t t  r  ; A    ˆ t t  r  ; B ˆ dd A  þ  ˆ dd B if   t  rl  and  t  rr   are associated to a peak height of 13 : 4% 8>><>>: whereˆ t t  r  ; A N ˆ t t  r  ; B ˆ t t  r  ; A  ¼  t  0  exp ˆ bb T  A  X    þ  1 h i ˆ t t  r  ; B  ¼  t  0  exp ˆ bb T  B  X    þ  1 h i ˆ dd A  ¼  t  0  exp ˆ bb T  r  ; A  X    þ  1 h i   t  0  exp ˆ bb T  l ; A  X    þ  1 h i ˆ dd B  ¼  t  0  exp ˆ bb T  r  ; B  X    þ  1 h i   t  0  exp ˆ bb T  l ; B  X    þ  1 h i We can proceed like this for any other chromatographicresponse:minimum resolution : ˆ  R R S ; min  ¼  min  pics  j  ; kj  ˆ  R R S ;  jk    run time : ˆ t t  r  ; max  ¼  max  peaks  j  ˆ t t  r  ;  j    symmetry factor : ˆ  I  I  sym  ¼ j ˆ t t  rr   þ  ˆ t t  rl    2ˆ t t  r  j ˆ dd maximum symmetry factor : ˆ  I  I  sym ; max  ¼  max  peaks  j  ˆ  I  I  sym ;  j    So this cascade of equations allows us to estimate each of these chromatographic responses in any point of theexperimental domain. This means that at this step, we areable to have a response surface for each chromatographicresponse. This is helpful to visualize the responses that areused to optimize the method. 2.3. Multi-criteria decision As most of the time, more than one response is used tooptimize the method of interest, we have to find acompromise between them. In other words, if an experimen-tal point is the best for a response, this point may be not the best for another response. The way to deal with this isassociatingDerringerfunctions[6]toeachoftheresponsesof interest and in taking the geometric mean (weighted if necessary), we obtain a desirability value for each experi-mental point.The Derringer function associated to a response  R  isdefined like this:  R a  A ; B ½    X    D a  0 ; 1 ½  Table 1The first three columns contain the Box–Behnken Experimental design (  X  1 ,  X  2  and  X  3 ) and the remaining columns contain the retention times  X  1  X  2  X  3  Peak 1 Peak 2 Peak 3 Peak 4 Peak 5 t  r   t  rl  t  rr   t  r   t  rl  t  rr   t  r   t  rl  t  rr   t  r   t  rl  t  rr   t  r   t  rl  t  rr  0   1   1 7.7 7.6 7.9 7.7 7.6 7.9 8.8 8.6 9.0 10.1 9.9 10.3 18.7 18.5 19.00   1 1 6.6 6.5 6.7 7.3 7.2 7.4 7.9 7.7 8.0 9.4 9.2 9.6 13.5 13.3 13.60 1   1 7.8 7.6 8.0 7.8 7.6 8.0 8.7 8.6 8.9 10.9 10.7 11.1 18.0 17.8 18.20 1 1 6.6 6.5 6.8 7.2 7.1 7.4 7.7 7.6 7.9 9.2 9.1 9.4 13.3 13.1 13.4  1 0   1 10.9 10.7 11.2 15.8 15.5 16.1 17.3 17.0 17.7 24.5 24.0 25.0 22.7 22.4 23.0  1 0 1 9.2 9.1 9.4 13.7 13.5 14.0 14.5 14.3 14.8 17.5 17.3 17.8 19.9 19.6 20.31 0   1 6.7 6.6 6.8 6.1 6.0 6.2 6.7 6.6 6.8 7.9 7.7 8.0 15.3 15.1 15.51 0 1 5.7 5.6 5.8 5.7 5.6 5.8 6.0 5.9 6.1 6.8 6.7 6.9 11.4 11.3 11.5  1   1 0 10.3 10.1 10.5 15.8 15.5 16.1 17.0 16.7 17.3 23.4 23.0 23.8 20.2 19.9 20.4  1 1 0 10.2 10.0 10.4 15.1 14.8 15.4 16.1 15.8 16.4 22.7 22.4 23.1 19.8 19.6 20.11   1 0 6.3 6.1 6.5 5.9 5.8 6.0 6.3 6.1 6.5 7.1 7.0 7.3 13.4 13.2 13.61 1 0 6.1 6.0 6.2 5.7 5.6 5.8 6.1 6.0 6.2 6.9 6.8 7.0 13.1 13.0 13.30 0 0 7.2 7.0 7.3 7.7 7.5 7.8 8.4 8.2 8.5 10.3 10.2 10.5 15.4 15.2 15.60 0 0 7.1 7.0 7.3 7.4 7.2 7.6 8.2 8.1 8.4 9.7 9.6 9.9 15.7 15.5 15.90 0 0 7.1 7.0 7.3 7.4 7.2 7.6 8.1 8.0 8.3 9.7 9.6 9.9 15.4 15.2 15.6 W. Dewe´ et al. / Chemometrics and Intelligent Laboratory Systems 74 (2004) 263–268  265  where 0 is associated to response values that are not satisfactory at all and 1 is associated to response values that are the most desirable.The overall desirability for   p  responses is calculated in anon-weighted case as following:  D 4  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  D 1    N      D  p  p p   a  0 ; 1 ½  2.4. Peak and chromatogram modeling  Each peak of the chromatogram is modeled by thefollowing function:  p ð t  Þ ¼ ( h exp    log  100 c   t     ˆ t t  r  ˆ t t  r     ˆ t t  rl  2   if   t  V ˆ t t  r  h exp    log  100 c   t     ˆ t t  r  ˆ t t  rr     ˆ t t  r   2   if   t  z ˆ t t  r  where  h  is the peak height and  c  is equal to 50 (if the left and right retention times are associated to 50% of the peak width) or to 13.4 (if the left and right retention times areassociated to 13.4% of the peak width).Although each part of this function is based on theGaussian function, it is dissymmetric with respect to theretention time at apex when the time intervals  t ˆ  r   t ˆ  rl (between the left retention time and the retention time at apex) and  t ˆ  rr   t ˆ  r   (between the right retention time and theretention time at apex) are different.If we consider the retention’s times at 50% of the peak height, then the function  p ( t  ) takes the values  h /2,  h  and  h /2when  t   is equal to  t ˆ  rl ,  t ˆ  r   and  t ˆ  rr  , respectively.As the peak area should be constant whatever thechromatographic conditions, the peak height is obtained by calculating the ratio between the geometric mean of the peak areas obtained across the experiments and thefollowing integration: Z   ˆ t t  r  0 exp "   log  100 c   t     ˆ t t  r  ˆ t t  r     ˆ t t  rl  2 # d t  þ Z   þ a ˆ t t  r  exp "   log  100 c   t     ˆ t t  r  ˆ t t  rr     ˆ t t  r   2 # d t  This integration can be calculated numerically usingappropriate software.The chromatogram is then predicted by summing thesefunctions  p ( t  ). 3. Example and discussion A chiral liquid chromatographic (LC) method wasdeveloped for the simultaneous determination of S-timolol,R-timolol and related substances. This method consisted inthe use of a Chiralcel OD-H column (250  4.6 mm; i.d.) packed with cellulose tris(3,5–dimethyl phenyl carbamate) Table 2The first three columns contain the Box–Behnken Experimental design (  X  1 ,  X  2  and  X  3 ) and the remaining columns contain the resolution between peaks2 and 3, the resolution between peaks 4 and 5, and the minimum resolution  X  1  X  2  X  3  RS23 RS45 Min RS Run time0   1   1 1.8 11.0 0.0 18.70   1 1 1.2 7.4 1.2 13.50 1   1 1.6 9.4 0.0 18.00 1 1 1.2 7.8 1.2 13.3  1 0   1 1.4 1.4 1.4 24.5  1 0 1 0.9 2.6 0.9 19.91 0   1 1.4 12.8 0.0 15.31 0 1 0.8 10.7 0.0 11.4  1   1 0 1.2 2.9 1.2 23.4  1 1 0 1.0 2.7 1.0 22.71   1 0 0.9 12.1 0.0 13.41 1 0 1.2 12.9 0.0 13.10 0 0 1.4 8.1 1.1 15.40 0 0 1.5 9.3 0.6 15.70 0 0 1.4 9.2 0.6 15.4Fig. 1. Desirability associated to the minimum resolution.Fig. 2. Overall desirability. W. Dewe´ et al. / Chemometrics and Intelligent Laboratory Systems 74 (2004) 263–268 266  coated on silica particules (5  A m) from Daicel LimitedIndustries (Tokyo, Japan). The mobile phase delivered at aflow rate of 1.0 ml min -1 was composed of a mixture of hexane, 2-propanol and DEA. The UV detection was performed at 297 nm. Other conditions are described inthe literature [7].The experimental domain of the three factors  X  1 ,  X  2 and  X  3  were [2%, 8%], [0.1%, 0.5%] and [15, 30  8 C],respectively. The codes (  1), (0) and (1) corresponding tothe lower, central and upper levels of each factor are usedfurther in the tables. In accordance with preliminarystudies [7], the experimental domain was selected in order to avoid operating conditions inappropriate for the LCsystem. The mobile phases containing concentrations of 2- propanol below 2% showed very long analysis times andthose containing higher concentrations than 8% showed anincrease of the column back pressure above the maximumtolerated by the chiral stationary phase for a long lifetime.The concentrations of diethylamine higher than 0.5%might deteriorate the chiral stationary phase and thereforecause a loss of efficiency. A competing effect of this aminehas been observed for concentrations between 0.1% and0.5%. Concerning the column temperature, values below15  8 C cannot be reached easily with the used LCequipment during routine analysis. Above 30  8 C, nosignificant changes in the elution order and retention timesof the peaks were noticed.A Box–Behnken experimental design [8] has beenfollowed to perform the experiments. It is an incompletefactorial design specifically selected for the optimization of quantitative factors. All design points are located at thecenter of the edges of t he hypercube, and are all situated onthe surface of a sphere [5]. In this case, star points out of the sphere as well as the axial points are avoided as they canlead to operating conditions inappropriate for the describedLC system. Moreover, with the Box–Behnken design, thenumber of experiments is reduced allowing to disposeenough time to equilibrate the chiral LC system betweeneach experiment and to execute each experiment induplicate. It is expected that the normal phase LC needsmore time for the equilibration than in reversed phase mode.The data are presented in Table 1. While performing the experiments, the mixture solution of the five analytes at different concentrations was injected so that each peak could be easily identified. However, when a coelution of peakswas observed, such as peak 1 and peak 2 in the 0/   1/   1experiment, the solutions of analytes were prepared indi-vidually and then injected. The different times could bemeasured. The responses that were used to assess the qualityof the method under the different conditions are: theminimum resolution, the resolution of the enantiomeric peak pair, the resolution between the two peaks of interest  Fig. 3. Chromatogram obtained in the optimal conditions.Fig. 4. Chromatogram predicted in the optimal conditions.Fig. 5. Observed versus predicted resolution of peaks 2 and 3.Fig. 6. Observed versus predicted resolution of peaks 4 and 5. W. Dewe´ et al. / Chemometrics and Intelligent Laboratory Systems 74 (2004) 263–268  267
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